On the Connection Between Lovelock Gravity and the Poincaré-Hopf Index

Abstract

This study explores a novel conceptual link between the dynamics of Lovelock gravity and the topological structure of spacetime. We introduce a theoretical framework in which the gravitational action—particularly in Lovelock theories is connected to the sum of topological indices of a vector field defined on the spacetime manifold, consistent with the Poincaré-Hopf theorem. Specifically, we propose that the value of the pth-order Lovelock action, recognized as a topological invariant in 2p dimensions, corresponds to the sum of indices associated with topological defects such as black hole horizons or spacetime singularities. Although the explicit construction of this vector field is left for future work, we present a foundational formulation and demonstrate its implications using the Schwarzschild solution, where the index sum reproduces the known Euler characteristic. This approach offers a topologically grounded perspective on gravitational dynamics and suggests new directions for understanding the geometric and physical properties of spacetime.